Mathematical justifi cation: the Babylonian example 371
Most explicit are some texts from late Old Babylonian Susa: TMS vii ,
TMS ix , TMS xvi. 22 Since TMS ix is closely related to the problem we have
just dealt with, whereas TMS vii investigates non-determinate linear prob-
lems and TMS xvi the transformation of linear equations, we shall begin
by discussing TMS ix ( Figures 11.3 and 11.4 ). It falls in three sections, of
which the fi rst two run as follows:
#1
- Th e surface and 1 length accumulated, 4[0 ́. ¿ 30, the length,? 20 ́ the width.] 23
- As 1 length to 10 ́ | the surface, has been appended,]
- or 1 (as) base to 20 ́, [the width, has been appended,]
- or 1°20 ́ [ ¿ is posited? ] to the width which together [ with the length ¿ holds? ] 40 ́
- or 1°20 ́ toge〈ther〉 with 30 ́ the length hol[ds], 40 ́ (is) [its] name.
- Since so, to 20 ́ the width, which is said to you,
- 1 is appended: 1°20 ́ you see. Out from here
- you ask. 40 ́ the surface, 1°20 ́ the width, the length what?
- [30 ́ the length. T]hus the procedure.
#2
- [Surface, length, and width accu]mulated, 1. By the Akkadian (method).
- [1 to the length append.] 1 to the width append. Since 1 to the length is
appended, - [1 to the width is app]ended, 1 and 1 make hold, 1 you see.
- [1 to the accumulation of length,] width and surface append, 2 you see.
- [To 20 ́ the width, 1 appe]nd, 1°20 ́. To 30 ́ the length, 1 append, 1°30 ́. 24
- [ ¿ Since? a surf ]ace, that of 1°20 ́ the width, that of 1°30 ́ the length,
- [ ¿ the length together with? the wi]dth, are made hold, what is its name?
- 2 the surface.
- Th us the Akkadian (method).
Section 1 explains how to deal with an equation stating that the sum of a
rectangular area ( l , w ) and the length l is given, referring to the situation
that the length is 30 ́ and the width 20 ́. Th ese numbers are used as identi-
fi ers, fulfi lling thus the same role as our letters l and w. Line 2 repeats the
22 A l l w e r e fi rst published by Bruins and Rutten 1961 who, however, did not understand their
character. Revised transliterations and translations as well as analyses can be found in H2002:
181–8, 89–95 and 85–9 (only part 1 ), respectively. A full treatment of TMS xvi is found in
Høyrup 1990 : 299–302.
23 As elsewhere, passages in plain square brackets are reconstructions of damaged passages that
can be considered certain; superscript and subscript square brackets indicate that only the
lower or upper part respectively of the signs close to that bracket is missing. Passages within
¿. . .? are reasonable reconstructions which however may not correspond to the exact
formulation that was once on the tablet.
24 My restitutions of lines 14–16 are somewhat tentative, even though the mathematical substance
is fairly well established by a parallel passage in lines 28–31.